One of the simplest models to trigger a strong first-order EWPT is to introduce a new bosonic degree of freedom to the scalar potential of SM.
Various Higgs portal models with (or without) a DM candidate have been widely studied 
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}.
The extension of SM with a complex singlet scalar for the above-mentioned purpose was proposed and studied in Refs. [1]}, [2]}, followed by extensive researches on its implications in subsequent works [2]}, [3]}, [12]}, [13]}.
In the Higgs-portal model with a real singlet scalar as the DM candidate, the scalar is stabilized by a \(\mathbb {Z}_2\)  symmetry and cannot have a vacuum expectation value (VEV) [40]}. DM phenomena related to this model have been studied in Refs. [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.
A previous study [49]} also shows that such a model cannot trigger a sufficiently strong first-order EWPT except when a large number of singlet scalars (\(\sim 10\) ) are introduced.
In the model with a global \(U(1)\)  group explicitly broken down to \(\mathbb {Z}_2\)  symmetry, Ref. [50]} shows that all phase transitions leading to the correct vacuum are of the second order. Furthermore,
Ref. [51]} shows that a large portion of the parameter space for the \(\mathbb {Z}_2\)  symmetry model will be ruled out by the required bubble nucleation condition, \(S(T_{n})/T_{n}\sim 140\) .
